Omnitruncated 7-simplex honeycomb
| Omnitruncated 7-simplex honeycomb | |
|---|---|
| (No image) | |
| Type | Uniform honeycomb |
| Family | Omnitruncated simplectic honeycomb |
| Schläfli symbol | {3[8]} |
| Coxeter–Dynkin diagrams |     
|
| 6-face types | t0123456{3,3,3,3,3,3} |
| Vertex figure |  Irr. 7-simplex |
| Symmetry | [math]\displaystyle{ {\tilde{A}}_8 }[/math]×16, [8[3[8]]] |
| Properties | vertex-transitive |
In seven-dimensional Euclidean geometry, the omnitruncated 7-simplex honeycomb is a space-filling tessellation (or honeycomb). It is composed entirely of omnitruncated 7-simplex facets.
The facets of all omnitruncated simplectic honeycombs are called permutahedra and can be positioned in n+1 space with integral coordinates, permutations of the whole numbers (0,1,..,n).
A7* lattice
The A*7 lattice (also called A87) is the union of eight A7 lattices, and has the vertex arrangement to the dual honeycomb of the omnitruncated 7-simplex honeycomb, and therefore the Voronoi cell of this lattice is an omnitruncated 7-simplex.
∪
∪
∪
∪
∪
∪
∪
= dual of .
Related polytopes and honeycombs
This honeycomb is one of 29 unique uniform honeycombs[1] constructed by the [math]\displaystyle{ {\tilde{A}}_7 }[/math] Coxeter group, grouped by their extended symmetry of rings within the regular octagon diagram:
| A7 honeycombs | ||||
|---|---|---|---|---|
| Octagon symmetry |
Extended symmetry |
Extended diagram |
Extended group |
Honeycombs |
a1
|
[3[8]] |
|
[math]\displaystyle{ {\tilde{A}}_7 }[/math] |
|
d2
|
<[3[8]]> |     
|
[math]\displaystyle{ {\tilde{A}}_7 }[/math]×21 |
|
p2
|
[[3[8]]] |  
|
[math]\displaystyle{ {\tilde{A}}_7 }[/math]×22 | |
d4
|
<2[3[8]]> |
|
[math]\displaystyle{ {\tilde{A}}_7 }[/math]×41 |
|
p4
|
[2[3[8]]] |
|
[math]\displaystyle{ {\tilde{A}}_7 }[/math]×42 |
|
d8
|
[4[3[8]]] | 
|
[math]\displaystyle{ {\tilde{A}}_7 }[/math]×8 |
|
r16
|
[8[3[8]]] |
|
[math]\displaystyle{ {\tilde{A}}_7 }[/math]×16 | 3
|
See also
Regular and uniform honeycombs in 7-space:
- 7-cubic honeycomb
- 7-demicubic honeycomb
- 7-simplex honeycomb
- Truncated 7-simplex honeycomb
- 331 honeycomb
Notes
- ↑ Weisstein, Eric W.. "Necklace". http://mathworld.wolfram.com/Necklace.html., OEIS sequence A000029 30-1 cases, skipping one with zero marks
References
- Norman Johnson Uniform Polytopes, Manuscript (1991)
- Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6 [1]
- (Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380-407, MR 2,10] (1.9 Uniform space-fillings)
- (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
Fundamental convex regular and uniform honeycombs in dimensions 2-9
| ||||||
|---|---|---|---|---|---|---|
| Space | Family | [math]\displaystyle{ {\tilde{A}}_{n-1} }[/math] | [math]\displaystyle{ {\tilde{C}}_{n-1} }[/math] | [math]\displaystyle{ {\tilde{B}}_{n-1} }[/math] | [math]\displaystyle{ {\tilde{D}}_{n-1} }[/math] | [math]\displaystyle{ {\tilde{G}}_2 }[/math] / [math]\displaystyle{ {\tilde{F}}_4 }[/math] / [math]\displaystyle{ {\tilde{E}}_{n-1} }[/math] |
| E2 | Uniform tiling | {3[3]} | δ3 | hδ3 | qδ3 | Hexagonal |
| E3 | Uniform convex honeycomb | {3[4]} | δ4 | hδ4 | qδ4 | |
| E4 | Uniform 4-honeycomb | {3[5]} | δ5 | hδ5 | qδ5 | 24-cell honeycomb |
| E5 | Uniform 5-honeycomb | {3[6]} | δ6 | hδ6 | qδ6 | |
| E6 | Uniform 6-honeycomb | {3[7]} | δ7 | hδ7 | qδ7 | 222 |
| E7 | Uniform 7-honeycomb | {3[8]} | δ8 | hδ8 | qδ8 | 133 • 331 |
| E8 | Uniform 8-honeycomb | {3[9]} | δ9 | hδ9 | qδ9 | 152 • 251 • 521 |
| E9 | Uniform 9-honeycomb | {3[10]} | δ10 | hδ10 | qδ10 | |
| En-1 | Uniform (n-1)-honeycomb | {3[n]} | δn | hδn | qδn | 1k2 • 2k1 • k21 |
 |  |
